Asymptotic expansion of 1-dimensional sub-manifolds of stable and unstable manifolds in a 4-dimensional symplectic mapping

We analytically compute asymptotic expansions of a 1-dimensional sub-manifold of stable and unstable manifolds in a 4-dimensional symplectic mapping by using the method called asymptotic expansions beyond all orders. This method enables us to capture exponentially small splitting of separatrices and also to obtain explicit functional approximations of the sub-manifolds. In addition, we show the condition with which homoclinic structure caused by crossing between the stable and unstable sub-manifolds is regarded as a direct product of 2-dimensional mappings.